Inheritors and Paradox Dorothy Grover The Journal of Philosophy, Vol. 74, No. 10, Seventy-Fourth Annual Meeting American Philosophical Association, Eastern Division. (Oct., 1977), pp. 590-604. Stable URL: http://links.jstor.org/sici?sici=0022-362X%28197710%2974%3A10%3C590%3AIAP%3E2.0.CO%3B2-D The Journal of Philosophy is currently published by Journal of Philosophy, Inc..
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THE JOURNAL OF PHILOSOPHY
INHERITORS AND PARADOX
*
T
HOSE who hold the traditional view that the primary role of the truth predicate is property ascription encounter problems when trying to determine the truth or falsity of sentences such as 'This is false' (the Liar sentence) which appear to be true if false, and false if true. I n order to save the truth property, it has been customary to seek ways of excluding such sentence5 from the extensions of 'true' and 'false'. The source of the problem, and also a way out of the seeming ascription of inconsistent properties, has been seen to lie sometimes with self-reference, sometimes with category mistakes, sometimes with natural language itself, and so the list goes on. But there is another avenue to explore. I n this paper I discuss the status of sentences such as the Liar from the point of view of a theory in which the use of 'true' and 'false' in English can be understood without supposing that these predicates ascribe predicates. On such a theory, since there is no ascription of properties, the Liar sentence cannot-at least for the reasons usually given-be paradoxical. The prosentential theory of truth is a theory of truth in which 'true' has a "prosentential" role. In " 'This is False' on the Prosentential Theory," I briefly explain the prosentential account of 'This is false', showing that whatever strangeness it does have can be found in many other sentences and also in terms involving "proforms." Because the prosentential account sketched there explains a number of intuitions we have concerning the Liar and because it is important to see how certain conceptual confusions have led to the construal of similar sentences and predicates as paradoxical and to the development of inconsistent formal languages, I extend that account here. In this paper I extend the account to cover sentences that are like the Liar but have different subject terms, strengthened versions of the Liar, and I also explain-from the point of view of the prosentential theory-why such sentences have been thought to be paradoxical. I n a later
* T o be presented in an APA symposium on Ungrounded Inheritors, December 30, 1977. John L. Pollock will comment; see this JOURNAL, this issue, 604406. I am indebted to Nuel Belnap, Charles Chastain, Susan Haack, Hide Ishiguro, and Robert L. Martin for their comments on earlier drafts. Also helpful were critical comments made in discussion of the paper when I read an earlier version at the University of Illinois at Chicago Circle and a t Wayne State University. 1See D. L. Grover, J. L. Camp, N. D. Belnap, "A Prosentential Theory of GCB Truth," Philoso#hical Studies, xxvrr, 2 (February 1975): 73-12.5-henceforth -for details. 2 Analysis, xxxv~,2 (January 1976): 80-83.
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paper I will cover the Heterological and Berry paradoxes and some of the paradoxes that have appeared in formal languages. I n that paper I will also show how a property ascribing truth predicate derives, without inconsistency, from a prosentential 'true'. I will not devote space in this paper to either motivating or explaining in detail the claim that 'true' and 'false' have prosentential roles in English. T h e reader is referred to GCB for that. With respect to certain points, reference will be made to sections of GCE. However, as a brief introduction to the theory, I will say a bit about what a prosentence is (GCB $2.1). Prosentences were introduced in my "Propositional Quantifiers," to provide natural-language readings of formulas containing bound propositional variables. Prosentences (also proverbs and proadjectives) were introduced by analogy with pronouns. Prosentences are like anaphoric pronouns in that they can be used both in the lazy way,4 as the pronoun 'he' is used in (1) John is tall a n d h e is heavy.
and qz~antificationally,4as 'it' is used in If a number is even, it is evenly divisible by 2.
Pronouns are different from prosentences in that pronouns occupy positions in sentences that singular terms can occupy; prosentences occupy positions in sentences that declarative sentences can occupy. Part of the claim that 'true' has a prosentential role is that it is used in forming prosentences, as in 'That is true', which is used in English as a prosentence of laziness (GCB $2.3). Just as 'he' in (1) has 'John' as its antecedent,e so 'That is true' in Mary: Chicago is large. John: If that is true, i t probably has a large airport.
has 'Chicago is large' as its antecedent. In each case the tie is grammatical. (In the case of 'That is true', although the tie derives from an anaphoric 'that', it is the whole of 'That is true' which is the prosentence.) Semantically, just as the referent of 'he' in (1) is the same as the referent of its antecedent 'John', so the propositional Journal of Philosophical Logic, I, 2 (April 1972): 111-136. Although I assume here a distinction between lazy and quantificational uses of pronouns, I don't mean to suggest that an adequate theory of pronouns must make the distinction. For the moment, the "distinction" is useful. 6 See B. H. Partee, "Opacity, Coreference, and Pronouns," Synthese, XXI, 3/4 (October 1970): 359-385, for a helpful discussion of pronouns. e T h e word 'antecedent' is a little misleading. Even though an antecedent usually occurs earlier in the discourse than a dependent proform, it doesn't have to. 3
4
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content (or simply "content") of 'That is true' is the same as that of its antecedent 'Chicago is large'. T h e prosentential accourit of 'That is true' captures features of 'true' which have hitherto been noticed but left unexplained. T h e connections that obtain between prosentences of laziness and their antecedents establish connections required in communication between what the speaker is saying and what someone else has said. When a pronoun of laziness is used, we know that in the simple cases no new object is being referred to, and, in the case of prosentences, that nothing new is being asserted or entertained. This last is a fact about 'true' that the Redundancy theorists drew attention to. Furthermore, an antecedent is explicitly acknowledged when a proform (i.e., a pronoun, prosentence, or proadjective, etc.; GCB 86) is used, and so it is made clear that the speaker is not plagiarizing. So also, as P. F. Strawson has pointed out, 'true' is used when expressing agreement, granting or considering a point. Proforms are also used to obtain generality. One way this is done is by means of the prosentence 'It is true'. Consider, for example, a generalized form of If the pope says that snow is white, snow is white.
Namely, If the pope says something, it is true.
W. V. Quine * has also pointed out that 'true' is needed when generalizing, but his account differs from the prosentential account, as on his account 'true' is metalinguistic. A prosentential 'true' is in the object language. Prosentences can be modified (GCB 99-101), as in 'That was true', 'That might be true', etc. Modified prosentences inherit as their content a modified form of their antecedent's content. I n most formal languages, connectives do the work of such verb modifications. 'That is false' is a modified prosentence which inherits as its content the contradictory of its antecedent's content. T h e prosentential account of 'This is false' goes something like this.9 Pronouns and prosentences of laziness, along with some other expressions in our language which I will mention later, acquire r "Truth," Proceedings of the Aristotelian Society, suppl. vol. xx~v (1950): 129-156. 8 Philosophy of Logic (Englewood Cliffs, N.J.: Prentice-Hall, 1970). sbtrawson, in "Truth," Analysis, xx, 6 (June 1949): 83-97, gives an account of the Liar in terms of his non-property-ascribing account of 'true'. T h e account of the Liar given here is very much in the spirit of what Strawson has to say.
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593
their referents or content-as the case may be-from antecedent expressions. I call such expressions "inheritors." If, for some reason, the antecedent either fails to denote, if it is a term, or fails to have propositional content, if it is a sentence, the inheritor fails to acquire an appropriate referent or content. There is such a failure in the case of 'This is false': as a modified prosentence it relies on its antecedent for content, but it is, unfortunately, its own antecedent and, as such, fails as an independent supplier of content. So 'This is false' lacks content. I.A. GROUNDED PRONOUNS
Pronouns of laziness are often used for stylistic reasons and perhaps also for lazy reasons, but, in addition, they (together with quantificational pronouns and other proforms) are useful semantically for the way in which they facilitate connections between one piece of a discourse and another. I n the simple cases, a pronoun of laziness has the same referent as its antecedent, and so, when such a pronoun is used, we know the same object is referred to. The connections can also be more complex: there are pronouns that resist treatment by coreferentiality theories (see Partee, op. cit.), and there are cases where a pronoun has as its antecedent an expression that belongs to a different grammatical category, as in John visited us. It was a surprise. No matter how complex the connections, however, the fact is that a pronoun of laziness, in acquiring a referent from another expres sion-or in "bringing" a referent forward-depends on another expression for the referent it has. And because of this, pronouns of laziness must be connected by the anaphoric relation to an expression that can supply a referent. Terms that can supply referents must acquire their referents in other ways: as proper names do, according to the causal theory of reference, or, as with demonstratives, by means of a demonstration. A term such as a pronoun of laziness which, in a given context, acquires its referent from another expression, will be called an inheritor. Terms such as proper names which acquire their referents in other ways (i.e., independently of another expression having a referent, or another expression having propositional content, as in the case 'John visited us') will be said to acquire their referents independently. A pronoun of laziness is grounded if it is connected to an expression that acquires its referent or propositional content independently, if at all. Note that groundedness does not guarantee a referent. Consider The King of France is bald, and he is wise.
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I.B. INHERITORS In the case of pronouns, the anaphoric relation locates the antecedent from which a referent is inherited, but there are limits to the connections that can be made by means of such grammatical ties. Where there are several candidate antecedents ambiguity can arise when a pronoun is used, and grammatical ties are not available for distant pieces of discourse. When a name is available it can often be used, but sometimes, in order to convey the fact that the same person or thing is being talked about, reference must be made in other ways. Consider Susan: I have a student in my class who says that his father works in the National Park. He has some good suggestions about places to visit. Tim: I don't know the student you are referring to, but I think he is someone Mary has in her class too. I'll get her to talk to him.
A definite description is used here to refer to someone who has been referred to in another place. T h e occasion I have in mind is one where the point for T i m is not to talk about Susan herself and what she was doing in using 'he', but to say something about the student Susan is talking about. Descriptive phrases such as 'just mentioned', 'been talking about', 'are referring to', which ostensibly describe discourse, may often be used merely to locate an antecedent piece of discourse from which a referent is inherited. What is sought is a way of making connections with other parts of the discourse so that it becomes clear-as it is with pronouns-that, say, the person talked about in this place is the same as the person talked about in that other place. Thus, insofar as each depends on another expression for its referent, Tim's 'the person you are referring to' and 'he' are on a par. Occurrences of definite descriptions which acquire their referents from other expressions are also inhe~itors.Like pronominal inheritors, an occurrence of a definite description inheritor is grounded if it is connected to an antecedent that can acquire a referent independently. I.C. UNGROUNDED INHERlTORS I n normal (perhaps all) circunlstances, successful communication requires that inheritors be grounded. However, in order to evaluate tile Liar, I must consider some circumstances in which inheritors are ungrounded. I consider cases where the grammatical picture is complete-that is, there is an antecedent wherever there ought to be one-but the antecedent fails to supply a referent. Ungroundedness can happen this way. A referring term purports
UNGROUNDED INHERITORS
595
to acquire its referent from another referring term, and this in turn purports to acquire its referent from yet another term, and things continue in this way, with each term a n inheritor, yet there is no term i n the sequence that acquires a referent independently. Such cases arise when there is either a circle or an infinite sequence of inheritors. Consider, for example, the problem of determining the referent of the underlined terms in (2) Mary: The person to refer to in your next sentence -you - are - going -. is someone I know.
Fay: She is not.
Given the "universal availability" of definite descriptions, circles can be very small, as in (3) T h e person I am now referring to is tall.
And, if we forget that definite descriptions are sometimes only inheritors, we may suspect inconsistency in (4) The oldest living person to whom I am not now referring is tall.
An infinite sequence can be generated as follows (5) So: The person referred to in S, is tall.
S,: The person referred to in S, is tall.
S,: The person referred to in S,,, is tall.
I n each of the above cases the definite descriptions purport to acquire referents from other (not necessarily distinct) expressions. Because there is no independent supplier, they fail to have referents. T h e result is a set of sentences with nondenoting subject terms. 1I.A. GROUNDED PROSENTENCES
Other proforms of laziness share with pronouns the feature that they inherit from antecedent expressions. Prosentences are no exception: 'That is true' has the same content as 'Snow is white' in (6) Tom: Snow is white.
Fay: That is true.
Thus prosentences are iizhe~ito~s, and so also are modified prosentences (GCB 99) which inherit a modified form of their antecedent's content. A sentence that does not inherit its propositional content from another must acquire it independently-in some such way, for example, as 'Snow is white' does. An occurrence of a prosentence is grounded if it is connected to an antecedent that acquires its content independently. 'That is true' is grounded in (6). (The distinctions outlined, both here and in section I, are made only intuitively.)
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THE JOURNAL O F PHILOSOPHY 1I.B. SENTENTIAL INHERITORS
Just as pronouns are not the only term inheritors, so also prosentences are not the only sentence inheritors: 'true' is used to form inheritors. I n the case of 'That is true' an anaphoric 'that' provides a grammatical tie between it and an antecedent sentence. If there is a desire to make connections with more distant pieces of discourse or if for some other reason 'that' won't do, then, as with the case of terms, another way must be employed. Consider, for example, the use of 'The first paragraph of this paper' in (7) If the first paragraph of this paper is true, then.
..
which might on some occasions of use be construed as locating the antecedent discourse from which content is inherited. T h e antecedent part of (7), construed as an inheritor, is in effect "standing in" for a conjunction of (roughly) the sentences of the first paragraph. [In section 111 I indicate treatments of (7) appropriate for other occasions of use.] Proper names can be used to locate antecedents, and so also can numerals, as in (8) Snow is white.
(9) (8) is true.
T h e antecedent can also be "displayed": (10) 'Snow is white' is true.
I n (10) the antecedent is a proper part of the inheritor. As inheritors with 'Snow is white' as antecedent, (9) and (10) are about whatever the sentence 'Snow is white' is about, and not about the sentence 'Snow is white'. I n all these cases t h e t r u t h predicate is t h e clue that we have a n inheritor o n o u r hands. T h e use of 'true' to form inheritors is clearly a natural extension of the prosentential 'true', even though such inheritors are not themselves prosentences. They are not prosentences because the ties they have with their antecedents are not grammatical (GCB 87). We need an anaphoric 'that' for that.10 T h e role in providing us with a sentence that can "stand in" for another is (at least a part of) what is in question here. Like prosentences, these inheritors can be modified, and should then be thought of as standing in for a modified form of the antecedent. A sentential inheritor is grounded just in case it is conl o In the examples discussed, 'that' in 'That is true' has always been anaphoric. However, an antecedent might sometimes be located by means of a demonstrative 'that'. In such a case, 'That is true', although still an inheritor, is not a prosentence. The same applies for 'This is true'.
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nected by an "inheritance chain" to an antecedent that can acquire its content independently. 1I.C. UNGROUNDED INHERITORS
Contexts similar to those in which we found ungrounded terms provide us with ungrounded sentences. The same kinds of circles and sequences are possible, with similar semantic problems. An example like (2) in which there is a content circle is (11) (a): (b) is true.
(b): That is true.
In this case (b) purports to acquire its content from (a), and (a) from (b). There is no independent supplier of content in the circle of antecedents. Very small circles such as we had in (3) are also possible: (12) This is true.
and, also, (13) (13) is true.
Sequences can be generated: (14) S,': S,' is true.
S,': S,' is true.
S,,': Sn+; is true.
I n this case each inheritor is standing for the next, but, because there is no supplier of content, all lack content. Modified forms of inheritors yield the familiar (15) (c): (d) is false.
(d): That is true.
(16) This is false.
(17) (17) is false.
And variations of the infinite sequence are also possible. As before, these inheritors all lack content-a consequence of their ungroundedness. The circles might be well called circles of discontent! 1II.A. PARADOXES
We have at last arrived at those sentences which undermine the thesis that 'true' and 'false' are, on every occasion, property-ascribing predicates. What is the prosentential alternative? As inheritors, the Liar and its related versions [e.g., (17)] depend on their ante-
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cedents for content. But these inheritors are ungrounded: they are not connected to antecedents that can acquire content independently. T h e problem of a simultaneous ascription of inconsistent properties doesn't arise: the "paradoxical" sentences lack content. Lack of propositional content is a problem, but it is not an isolated problem that only "paradoxical" sentences have: sentences such as (11) lack content, in similar circuinstances terms lack referents, and I will show in section VI that the same problem can arise with definitions. 1II.B. GENERALIZED VERSIONS OF THE LIAR
I have discussed so far only "lazy inheritor" versions of the Liar. There are also generalized versions like (18) Anything that is identical with this sentence is false.
Quantificational uses of 'true' were explained in GCB by means of translations into a language containing bound propositional variables, and then natural-language paraphrases were provided in English * (GCB 91-95) using 'It is true' as a quantificational prosentence capturing the bound variables. Thus 'Everything John says is true' was translated as vp(If John says that p, then p) il: the point was not just to assert what John had asserted, but to i ntroduced explicitly tie it to John's saying it. T h e conditional '1, by Nuel Belnap l1 in his analysis of conditional assertion, was used when there was no desire to say anything about John. Thus vp(John says that p/p) Similar analyses can be given of (7), and are appropriate when, for example, the speaker doesn't actually know what has been said in the first paragraph. So, following the program of English analyses suggested in GCB, (18) can be rendered-making the modified quantificational prosentence explicit-as (18') Anything that is identical with this sentence is such that it is false.
T h e propositional content that a quantificational prosentence contributes to the sentence it occurs in, comes from the possible contents of the prosentence's substituends (sentences that yield substitution instances 12). Often the propositional content of the whole sentence can be determined if.we know only what the possible contents of the prosentence's substituends are, even though we don't 11 "Restricted Quantification and Conditional Assertion," in H. Leblanc, ed., Truth, Syntax and Modality (Amsterdam: North-Holland, 1973). 12 See my "Propositional Quantifiers," op. cit., pp. 116/7.
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know which contents the substituends actually have. I n these cases the problem of ungroundedness does not arise. A problem can arise, however, when, in determining the content of the whole, the specific content of a given substituend must first be determined. Insofar as quantificational prosentences contribute content acquired from substituends, they are inheritors. A substituend is crucial if excluding it from the set of the prosentence's substituends would alter the propositional content-if any-of the whole sentence.lS Clearly, a quantified prosentence is ungrounded if it has a crucial substituend that is ungrounded. I t is also ungrounded if the crucial substituend is the quantified sentence: the prosentence purports to acquire content from the quantified sentence and the quantified sentence from the prosentence. This is the case with (18'). T h e modified prosentence occurring in (18') has (18') as a substituend. T h e substituend is crucial since it is the only sentence identical with (IS'), and so the sentence whose content (if it had any) would be denied by means of 'It is false'. So (18') lacks content-it is not inconsis tent. IV. GROUNDED/UNGROUNDED
T h e grounded/ungrounded distinction separates those occurrences of inheritors which have content from those which don't. Included in the former are grounded occurrences of inheritors like (9) and (lo), and included in the latter are ungrounded inheritors such as (11)-(17). T h e set of occurrences of sentences that are useful in communication includes those which acquire content independently (e.g., 'Snow is white') and grounded inheritors. I t is interesting to note that this set corresponds, roughly, to that set of sentences to which Kripke l 4 has assigned a truth value at tlie smallest fixed point, and of which he says,
...the ,
smallest fixed point is probably the natural model for the intuitive concept of truth (708).
Presumably he has in mind a concept having to do with a truth property, but never mind that. There is agreement that the grounded inheritors are the inheritors we care about: they are the ones we use for successful comn~unication. There is another respect in wllicll there is agreement between 1s 'Crucial' has to be extended to cover cases where a family of substituends is crucial. And various alternatives must be considered, like those which arise in section VI, before the grounded/ungrounded distinction can be made clearly here. LXXII,19 (Nov. 6, 1975): l4"Outline of a Theory of Truth," this JOURNAL, 690-7 16. A similar model is presented by R. L. Martin and P. ItT.Woodruff, "On Representing 'True-in-L' in L," Philosophia, v, 3 (July 1975): 213-217.
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this prosentential account and Kripke's theory. Consideration of examples like Dean: Everything Nixon said about Watergate is false. Nixon: Everything Dean said about Watergate is false.
led him to the following moral:
.. .an adequate theory must allow our statements involving the notion of truth to be risky: they risk being paradoxical if the empirical facts are extremely (and unexpectedly) unfavorable (692). Because the context of an occurrence of a sentential inheritor determines what content it has (or what it contributes to a quantified sentence), and thereby, whether it is grounded, empirical facts do have a role, on my account, in determining whether an utterance is risky-as they should. Kripke goes on to say, however, There can be no syntactic o r semantic "sieve" that will winnow out the "bad" cases while preserving the "good" ones (692).
T h e cases Kripke has in mind are sentences. Because the content of an inheritor changes from one context to another, my cases must in contrast be utterances. On my account, the good cases are separated from the bad whenever an analysis of the content of a sentential inheritor is given. Sensitive to the effects of context, this (informally developed) account provides a sieve. V. A CAUSE OF "PARADOX"
I have defined an inheritor as an expression that acquires either its referent or its content from another expression, and I've claimed that 'true' is used in the construction of sentential inheritors. Suppose a sentential inheritor is not recognized as such. That is, it is not recognized that its content is acquired from another expression. In that case it will be assumed that the content that it has is acquired independently. In languages like English, sentential inheritors can be connected to (and so purport to inherit from) any sentence, including themselves-that is, sentential inheritors are "universally available." Also, in such languages, inheritors can be modified so that they inherit the contradictory of their antecedent's content. Then inconsistencies arise if content is also acquired by inheritors in some way independently. Suppose, for example, it is not thought that 'true' is used in inheritors, but that both 'true' and 'false' are property-ascribing predicates. Then sentences containing 'true' and 'false' are thought to acquire their content independently, and not from other expressions. That might be all right on
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its own, but it is also generally assumed that (roughly) ' X is true' can be replaced by X, and ' X is false' by the contradictory of X. This fits with a prosentential 'true', but combined (without restriction) with the other, it leads to a problem. 1'11 try to give a picture of what is going on. Suppose a is a sentential inheritor, and suppose, also, that we write as a suffix the antecedent a has in a particular context. Thus the case where a has 'snow is white' as its antecedent is represented by a'8now is white'
Given this way of representing the situation, one easily envisages the possibility of combinations like (19) (aha
jb)a-,
( c ) ~ %
(dIma-a
So far as the prosententialist is concerned, none of the occurrences of a is connected to an expression that can acquire content independently, and so, in each case, the inheritor is ungroundedmore circles of discontent. So a fails to have content. Suppose, on the other hand, the role of the inheritor has not been recognized, and it is assumed that a has propositional content. If, in addition, the inheritor is still cashed in for its antecedent, there is a problem. No matter what content a is assumed to have, (19 b) and (19 d) are inconsistent: they both allow the simultaneous affirmation and denial of a. I n the case of (19 b), for example, we have it expressing the proposition that a. But a also acquires content from -a, so a is cashed in for -a. So we then have (19 b) expressing the proposition that wa!T h e Liar sentence has the form of (19 d). Because it is impossible for universally available modifiable inheritors to consistently acquire content independently, inheritors must not go unrecognized. These inheritance problems arise for definitions also. Since in a definition the definienclum acquires its content from the definiens, that occurrence of the definiendum is an inheritor. I n this case the problems are well recognized. The definiendum must not occur in the definiens-otherwise the definiendum is ungrounded. I t is also realized that there are cases such as H=,,zH where, were the "definition" taken seriously, an "inconsistency" arises. But of course H hasn't really been defined, since the expression it is supposed to acquire content from does not have content independently.
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THE jOURNAL OF PHILOSOPHY VI. A STRENGTHENED VERSION OF THE LIAR
Are inheritors-properly understood-free from inconsistency? Obviously I have no proof that they are, but I can show that some problems that arise for property-ascribing theories do not arise on this account. Consider, first, a version of the Liar sentence. Let X be 'X is false'. Using (syntactical) inheritance properties, it seems that if X is false, X is true, and that if X is true, X is false. These have the "form" of contradictories, but there is no contradiction because each lacks content. I t might be thought that talk of ungroundedness and lack of content lands this account in the class of third-value "solutions." If it does, it faces the usual objection: paradox has not been eliminated; it has been relocated. Consider (20) (20) is false, or (20) is ungrounded.
According to the property-ascribing view, if (20) has one of the mutually exclusive properties true, false, and ungrounded, it aiso has another of those properties: thus, if (20) is true, it is also either false or ungrounded1 Because 'ungrounded' does not introduce a third property, this problem does not arise for a prosentential 'true'. '(20) is true' and '(20) is false' are only inheritors. There is no conflict between '(20) is true' and (20) because '(20) is true' has the same content as (20): to suppose that (20) is true just is to suppose that either (20) is false or (20) is ungrounded. T h e part of (20) that supplies content is '(20) is ungrounded', so that could be the only source of inconsistency. I will use the three-valued Weak and Strong valuations of S. C. Kleene l5 to picture what is going on. As 'X is true' is an inheritor, the value of 'X is true' should be the same as that of X, and the value of 'X is false' the complement of the value of X. The grounded/ungrounded distinction has been made so far only for inheritors with simple antecedents. It will have to be extended to cover sentences like (20) and inheritors with such mixed antecedents. I consider only conjunctions and disjunctions. T h e point of the grounded/ungrounded distinction was to separate occurrences of inheritors which have content from those which don't. I n the case of sentences like (20) one might decide-depending, presumably, on where the distinction is to be used-either that a sentence is ungrounded if its content cannot be completely determined or that it is ungrounded if no content at all can be determined. 16 Introduction
to ,?fetamatheinati~s(Nerv York: Van Xostrand, 1952).
UNGROUNDED INHERITORS
603
T o begin with I will say that an occurrence of an inheritor is grounded, if it is connected to an antecedent with conjunctive (disjunctive) parts that can acquire content independently, or, if any of those parts are inheritors, then those inheritors must be grounded,; a conjunction (disjunction) is grounded, if all its conjuncts (disjuncts) that are inheritors are grounded,. Consider now (20,) (20,) is false, or (20J is ungrounded,.
According to the definition, (ZO,), '(20,) is true', and '(20,) is false' are ungrounded,. So '(20,) is ungrounded,' is assigned the value t. When using T..\Teak Kleene, ungrounded, inheritors will be assigned u (are undefined); on Strong Kleene, an ungrounded, inheritor has the value u unless its antecedent has a conjunct with value f, or a disjunct with value t. I n these cases an inheritor either inherits those values, or, if it is modified by 'false', it has the values t and f, respectively. I believe (ZO,), '(20,) is true', and '(20,) is false' are assigned unique values on each of these valuations. Weak Kleene. Since '(20,) is true' and '(20,) is false' are ungrounded,; they have the value u. Since one disjunct of (20,) has the value u, (20,) has the value u. Strong Kleene. Because '(20,) is ungrounded,' has the value t, (20,) and '(20,) is true' do too; '(20,) is false' will then have the value f. T h e second alternative is to say that an inheritor is ungroundedp if it is not connected to an antecedent with a conjunct (disjunct) that can acquire content independently; a ccnjunction (disjunction) is ungrounded, if it contains only ungrounded, inheritors. Consider now (20,) (20,) is false, or (20,) is ungrounded,.
(20,) is grounded, so '(20,) is grounded,' has the value f. O n both Weak and Strong Kleene (20,), '(20,) is true' and '(20,) is false' have the value u. A third "alternative" is to say that a sentence is grounded, if it is either true or false. But then 'grounded,' is a disguised inheritor. If X has no content, 'X is true' and 'X is false' don't either. So 'X is grounded' will lack content also. If 'true' and 'false' are to be used in definitions, care must be taken to see that the inheritors they occur in are grounded. VII. SEMANTIC AND LOGICAL PARADOXES
If this account of the Liar is successful in the way I think it is, the distinction between the semantic and logical paradoxes collapses.
604
1 H E JOUHNAI, OF I'HILOSOPHY
I believe that in each case the problems stem from ungrounded modified inheritors. Formal-language variables are very like quantificational proforms: the ungrounding problems that arise for inheritors are problems that variables can have also. DOROTHY GROVER
University of Illinois at Chicago Circle
T H E LIAR STRIKES BACK *
I
N her very interesting paper, Professor Grover has used the prosentential theory of truth to motivate a solution to the liar paradox. However, I will urge, first, that the solution she has offered really has nothing to do with the prosentential theory of truth, and second, that the solution ultimately breaks down for the most interesting versions of the paradox. T h e prosentential theory of truth has two parts: a positive account of the way 'true' functions in English, and the negative claim that 'true' does not function as a property-ascribing predicate. I am inclined to accept the positive account but deny the negative claim. The basis for the latter denial is that by using only locutions sanctioned by the prosentential theory, we can construct a property-ascribing truth predicate for propositions: PisTRUE
= (3s)
( 3 C ) (S is a sentence a n d C a set of circumstances under which a n utterance of S would express P , a n d for each proposition, if anyone said that it is true by uttering S under circumstances C then it is true).
Thus the negative thesis is indefensible. Let us reject it and identify the prosentential theory with its positive account. So construed, the prosentential theory is perfectly compatible with other accounts of the way 'true' works in English. I n particular, it is compatible with an account that takes 'that is true' to ascribe truth to a proposition referred .to by the pronoun 'that'. I propose that these are equally correct accounts of the same phenomena. There is no basis for saying that one account must be correct and the other incorrect. Turning to the liar paradox, it is my contention that most contemporary discussions of it suffer from a rather gross confusion. Abstract of a paper to be presented in an APA symposium on Ungrounded Inheritors December 30, 1977, commenting on a paper by Dorothy Grover; see this JOURNAL, this issue, 590-604.
